Multimodal authorization method, system and device

ABSTRACT

The invention includes methods, systems and devices for authenticating transactions. A method may begin by enrolling at least two biometric specimens to a database. A false acceptance ratio (“FAR”) is determined for each of the specimens, and authorization options are identified. A cost value is calculated for at least one of the options to provide a calculated cost value (“CCV”). The CCV may be a function of the FAR(s) of the specimen(s) corresponding to the option. An acceptable cost value range (“ACV range”) may be identified, and compared to the CCV. If it is determined that the CCV is in the ACV range, then the option is selected. If the CCV is in the ACV range, then a set of biometric samples is provided (the “sample set”). The sample set is compared to the biometric specimens, and it is determined whether the biometric samples match the biometric specimens. If the biometric samples match the biometric specimens, then the transaction is authorized.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority to U.S. provisionalpatent application Ser. No. 60/685,429, filed on May 27, 2005. Thisapplication is a continuation-in-part and claims the benefit of U.S.patent application Ser. No. 11/273,824 filed on Nov. 15, 2005, whichclaims priority to U.S. provisional patent application No. 60/643,853,which was filed on Jan. 14, 2005.

FIELD OF THE INVENTION

The present invention relates to authorization systems, such as thosethat are used to authorize a purchase transaction.

BACKGROUND OF THE INVENTION

Reliable personal authentication is becoming increasingly important. Forexample, transactions that rely man-made personalized tokens or otheridentifying instruments to verify the identity of an individual arebeing targeted by criminals to engage in identity theft. Traditionalsecurity measures rely on knowledge-based approaches such as passwordsand PINs or on token-based approaches such as swipe cards and photoidentification to establish the identity of individuals. Despite beingwidely used, these are not very secure forms of identification. It isestimated that hundreds of millions of dollars are lost annually incredit card fraud in the United States due to consumer misidentificationand identity theft. Biometrics offers a reliable alternative. Biometricsis the method of identifying an individual based on his or herphysiological and/or behavioral characteristics. Examples of biometricsinclude fingerprint, face recognition, hand geometry, voice, iris, andsignature verification. Biometrics may be used for immigration, airport,border, and homeland security. Wide scale deployments of biometricapplications such as the US-VISIT program are already being done in theUnited States and elsewhere in the world.

Despite advances in biometric identification systems, several obstacleshave hindered their deployment. For example, for every biometricmodality there may be some users who have illegible biometrics. Forexample a recent NIST (National Institute of Standards and Technology)study indicates that approximately 2 to 5% of the population does nothave legible fingerprints. Such users would be rejected by a biometricfingerprint identification system during enrollment and verification.Handling such exceptions is time consuming and costly, especially inhigh volume scenarios such as in retail stores where thousands oftransactions may be subject to authentication each day. Using multiplebiometrics to authenticate an individual will alleviate this problem;and, using multiple biometrics in a data fusion logic process mayachieve a quicker means of acquiring an accurate identification match.

Furthermore, unlike password or PIN based systems, biometric systemsinherently yield probabilistic results and are therefore not fullyaccurate. In effect, a certain percentage of the genuine users will berejected (false non-match) and a certain percentage of impostors will beaccepted (false match) by the system.

SUMMARY OF THE INVENTION

The invention includes a method of authorizing a transaction. In onesuch method, at least two biometric specimens are enrolled. A first oneof the specimens is a first type, and a second one of the specimens is asecond type. A false acceptance ratio (“FAR”) is determined for each ofthe specimens. Authorization options are identified, each optionrequiring a match to one or more of the biometric specimens. A costvalue is calculated for at least one of the options to provide acalculated cost value (“CCV”). The CCV may be a function of the FAR(s)of the specimen(s) corresponding to the option.

An acceptable cost value range (“ACV range”) may be identified, andcompared to the CCV. If it is determined that the CCV is in the ACVrange, then the option is selected. If it is determined that the CCV isnot in the ACV range, then the option may be discarded for use inauthorizing that transaction.

If the CCV is in the ACV range, then a set of biometric samples isprovided (the “sample set”). The sample set has biometric samples of thesame types as those corresponding to the selected option. The biometricsamples are compared to the biometric specimens, and it is determinedwhether the biometric samples match the biometric specimens. If thebiometric samples match the biometric specimens, then the transaction isauthorized.

The invention may be embodied as a computer readable memory device or asystem which is capable of carrying out methods according to theinvention. In one such system, there is (a) a computer capable ofexecuting computer-readable instructions, (b) at least one biometricspecimen reader in communication with the computer, (c) at least onebiometric sample reader in communication with the computer, (d) adatabase, and (e) computer-readable instructions provided to thecomputer for causing the computer to execute certain functions. Theinstructions may be stored on a memory device and provided from thememory device to the computer. Those functions may include (i) enrollingin the database at least two biometric specimens via one or more of thebiometric specimen readers, a first one of the specimens being a firsttype, and a second one of the specimens being a second type, (ii)determining a false acceptance ratio (“FAR”) for each of the specimens,(iii) identifying authorization options, each option requiring a matchto one or more of the biometric specimens (iv) calculating a cost valuefor at least one of the options to provide a calculated cost value(“CCV”), wherein the CCV is a function of the FAR(s) of the specimen(s)corresponding to the option (v) identifying an acceptable cost valuerange (“ACV range”), (vi) comparing the CCV to the ACV range, (vii)determining whether the CCV is in the ACV range, (viii) selecting theoption if the CCV is in the ACV range, (ix) providing a set of biometricsample(s) (the “sample set”) via one or more of the biometric samplereaders, the sample set having biometric sample(s) of the same type(s)as those corresponding to the selected option (x) comparing thebiometric sample(s) to the biometric specimen(s) (xi) determiningwhether the biometric sample(s) match the biometric specimen(s) (xi) ifthe biometric sample(s) match the biometric specimen(s), thenauthorizing the transaction.

The false acceptance ratios may be determined using the followingequation

FAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au))f(x|Im) 𝕕x.The cost values may be determined using the following equation

${Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}\;{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}\;{{FAR}_{i}.}}}}$

BRIEF DESCRIPTION OF THE DRAWINGS

For a fuller understanding of the nature and objects of the invention,reference should be made to the accompanying drawings and the subsequentdescription. Briefly, the drawings are:

FIG. 1, which is a schematic diagram of a bimodal biometricpoint-of-sale system according to the invention. This system usesbiometric authentication to authorize a purchase/sale transaction thatis being validated by the use of fingerprint and signature biometrics.

FIG. 2, which is a schematic diagram of another biometric point-of-salesystem according to the invention. In this system, it is possible to useseveral biometric modalities to validate the identification of someoneusing the system.

FIG. 3 illustrates a method according to the invention.

FIG. 4 illustrates a method according to the invention.

FIG. 5, which illustrates certain aspects of a device and a systemaccording to the invention;

FIG. 6, which represents example PDFs for three biometrics data sets;

FIG. 7, which represents a joint PDF for two systems;

FIG. 8, which is a plot of the receiver operating curves (ROC) for threebiometrics;

FIG. 9, which is a plot of the two-dimensional fusions of the threebiometrics taken two at a time versus the single systems;

FIG. 10, which is a plot the fusion of all three biometric systems;

FIG. 11, which is a plot of the scores versus fingerprint densities;

FIG. 12, which is a plot of the scores versus signature densities;

FIG. 13, which is a plot of the scores versus facial recognitiondensities;

FIG. 14, which is the ROC for the three individual biometric systems;

FIG. 15, which is the two-dimensional ROC—Histogram_interpolations forthe three biometrics singly and taken two at a time;

FIG. 16, which is the two-dimensional ROC similar to FIG. 15 but usingthe Parzen Window method;

FIG. 17, which is the three-dimensional ROC similar to FIG. 15 andinterpolation;

FIG. 18, which is the three-dimensional ROC similar to FIG. 17 but usingthe Parzen Window method;

FIG. 19, which is a cost function for determining the minimum cost givena desired FAR and FRR;

FIG. 20, which depicts a method according to the invention; and

FIG. 21, which depicts another method according to the invention.

FURTHER DESCRIPTION OF THE INVENTION

FIGS. 1 and 2 illustrate generally systems 10 that may be used to carryout the invention. In these figures there are shown biometric inputdevices 13, which may be used to provide biometric samples. A networkserver 16 may provide the samples to a computer 19. The computer 19 maybe in communication with a database 22 of biometric specimens, and thecomputer 19 may be able to determine whether the biometric samplesprovided at a point-of-sale terminal 25 match biometric specimens storedin the database 22.

FIG. 3 illustrates a generally a method that may be used to carry outthe invention. In FIG. 3 a buyer registers 50 biometric specimens usingone or more biometric readers. The specimens are provided 53 to in adatabase for later use. When a buyer desires to purchase goods orservices from a seller, the buyer provides 56 biometric samplescorresponding to the biometric specimens that were previously stored inthe database. The computer then compares 59 the specimens to the samplesto determine 62 whether there is a match between the set of samples andthe set of specimens. If a match is determined 62, the transaction isauthorized 65, and funds may be transferred 68 from the buyer'sfinancial institution to the seller's financial institution.

FIG. 4 illustrates in more detail how such a transaction may beauthorized. Although FIG. 3 illustrates the invention with regard to apurchase/sale transaction, the invention is not limited to such atransaction, and so FIG. 4 illustrates a method according to theinvention in general terms. The invention may be implemented as a methodexecuted by an authorization system, such as a facility-access system ora point-of-sale system. The authentication system may be used todetermine whether a person should be allowed access to a facility, orwhether a financial transaction should be allowed to proceed. Forexample, the financial transaction may be the purchase of goods orservices using a credit account or a debit account maintained by afinancial institution such as a bank.

According to the invention, biometric identification may be accomplishedusing multiple biometric types. For example, the types of biometricsthat may be used include fingerprints, facial images, retinal images,iris scans, hand geometry scans, voice prints, signatures, signaturegaits, keystroke tempos, blood vessel patterns, palm prints, skincomposition spectrums, lip shape, and ear shape.

In one such method or authorizing a transaction, at least two biometricspecimens are enrolled 100. A first one of the specimens may be a firsttype, and a second one of the specimens may be a second type. Forexample, the specimens may be from different body parts. For example,one of the specimens may be a fingerprint and another of the specimensmay be an iris scan. Or, one of the specimens may be a fingerprint froma person's thumb, and one of the specimens may be a fingerprint from theperson's index finger. The biometric specimens may be saved in adatabase for later use. A false acceptance ratio (“FAR”) may bedetermined 103 for each of the specimens in the database. Determining103 an FAR may be done in a number of ways, but a particular way wouldbe to determine 103 false acceptance ratios using the followingequation;

FAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au))f(x|Im) 𝕕x.Details corresponding to this equation are given later in this document,

Authorization options may be identified 106. Each option may require amatch to one or more of the biometric specimens in order for atransaction to be authorized. For example, if three biometric specimens(S1, S2 and S3) are enrolled 100 in the database, the followingauthorization options may be identified 106:

Option 1: use S1 by itself to authorize a transaction,

Option 2: use S2 by itself to authorize a transaction,

Option 3: use S1 and S2 together to authorize a transaction,

Option 4: use S2 and S3 together to authorize a transaction,

Option 5: use S1 and S3 together to authorize a transaction, and

Option 6: use S1, S2 and S3 together to authorize a transaction.

It will be noted that the possible options may correspond to anarrangement of all or part of the set of biometric specimens. As such,an authorization option may correspond to a permutation of the set ofbiometric specimens.

To illustrate the concept, consider that if Option 1 is selected, thenin order to authorize a transaction, a person would need to provide abiometric sample of the same type as specimen S1, and the sample wouldneed to match specimen S1. There are many ways to determine matchesbetween specimens and samples, and those will not be detailed herein.

As another example, if Option 4 is selected, then in order to authorizea transaction, a person would need to provide a first biometric sampleof the same type as specimen S2, as well as a second biometric sample ofthe same type as specimen S3. In addition, the first sample would needto match S2 and the second sample would need to match S3.

A cost value may be calculated 109 for at least one of the options toprovide a calculated cost value (“CCV”). The CCV may be a function ofthe specimen FAR or specimen FARs (as the case may be), which correspondto the option. Calculating 109 a CCV may be done in a number of ways,but a particular way would be to calculate the CCV using the followingequation:

${Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}\;{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}\;{{FAR}_{i}.}}}}$Details corresponding to this equation are given later in this document.

An acceptable cost value range (“ACV range”) may be identified 112 andthe CCV for the option may be compared 112 to the ACV range. The ACV maybe identified based on experience. For example, it may be known fromexperience that requiring an ACV of a particular value will produce anindication as to whether a match exists between specimens and samples ina reasonable amount of time and with an acceptable number of falseacceptances. If so, then a system administrator may identify 112 andprovide the ACV range. The CCV may be compared 115 to the ACV range, anda determination may be made 118 as to whether the CCV is within the ACV.

If the CCV is in the ACV range, the option corresponding to the CCV maybe selected 121, and a request for a set of biometric samples may bemade. Upon receiving biometric samples of the same type required by theselected option the samples may be provided 124 and the specimenscorresponding to the selected option may be compared 127 to the samples.A determination 130 may be made as to whether each specimen matches oneof the samples. If the specimens and samples are determined 130 tomatch, then the transaction may be authorized 133.

For example, if the cost values of the six options identified above are:

-   -   CV1=1.2    -   CV2=2.2    -   CV3=0.3    -   CV4=0.9    -   CV5=5.0    -   CV6=3.4        and if the ACV range is selected to be cost values that are less        than 2.1, then Options 1, 3 and 4 would be identified as        acceptable because they each have a CCV that is less than the        ACV. Options 2, 5 and 6 would be deemed unacceptable because        they each have a CCV that is above 2.1. Of those options that        are identified as acceptable, one of the options may be selected        121, for example, Option 3 may be selected 121 because it has        the lowest cost value. The user would then be prompted to        provide samples according to Option 3, that is one sample of the        same type as S1 and one sample of the same type as S2. If S1 is        a fingerprint and S2 is an iris scan, the user would be prompted        to provide a fingerprint sample and an iris scan sample, and        these samples would then be provided 121 to a matching process        in order to determine 130 whether a match exists between S1 and        the fingerprint sample, and also whether a match exists between        S2 and the iris scan sample. If a match for the fingerprint and        a match for the iris scan are determined 130, then the        transaction would be authorized 133. For example, if the        transaction is authorized 133, the buyer that supplied the        samples would be allowed to purchase goods on credit from a        vendor.

FIG. 5 illustrates a device 200 and a system 201 according to theinvention. The invention may be implemented as a computer readablememory device 200 having stored thereon instructions 203 that areexecutable by a computer 206 that is part of an authorization system201, which may be used to authorize a transaction. For example, thecomputer readable memory device 200 may be read-only-memory, such as acompact disc. The memory device 200 may have stored thereon instructionsfor carrying out a method that is in keeping with the descriptionprovided above. For example, the instructions may be executable by thecomputer 206 to cause the computer 206 to (a) enroll into a database 209at least two biometric specimens, a first one of the specimens being afirst type, and a second one of the specimens being a second type, (b)determine an FAR for each of the specimens, (c) identify authorizationoptions, each option requiring a match to one or more of the biometricspecimens, (d) calculate a CCV for at least one of the options, whereinthe CCV is a function of the FAR(s) of the specimen(s) corresponding tothe option, (e) identify an ACV, (f) compare the CCV to the ACV range,(g) determine whether the CCV is in the ACV range (h) select the optionif the CCV is in the ACV range (i) receive and provide a set ofbiometric sample(s) (the “sample set”) corresponding to the specimentype(s) of the selected option (j) compare the biometric sample(s) tothe biometric specimen(s), (k) determine whether the biometric sample(s)match the biometric specimen(s), and (l) if the biometric sample(s)match the biometric specimen(s), then authorize the transaction.Biometric specimen readers 212 and biometric sample readers 215 mayinterface with the computer 206 for the purpose of obtaining biometricspecimens and biometric samples, respectively, and providing thespecimens and samples to the computer 206. The equations for FAR and CCVidentified above may be used in the instructions to the computer 206.

It should be noted that the invention may be used to provide tokened ortokenless authorization of commercial transactions between a buyer andseller using a computer network system and multiple biometrics. Such asystem may or may not also include a token of some type, such as acredit card, driver's license, identification instrument or the like.Such a system may also include a PIN or other indexing device to speedthe authentication process.

Having identified two equations that may be used in the invention,further details of those equations are now provided. The FAR and CCVequations are part of a larger discussion about biometric fusion, whichis the study of how two or more biometric types (sometimes referred toas modalities) may be used. Algorithms may be used to combineinformation from two or more biometric modalities. The combinedinformation may allow for more reliable and more accurate identificationof an individual than is possible with systems based on a singlebiometric modality. The combination of information from more than onebiometric modality is sometimes referred to herein as “biometricfusion”.

Reliable personal authentication is becoming increasingly important. Theability to accurately and quickly identify an individual is important toimmigration, law enforcement, computer use, and financial transactions.Traditional security measures rely on knowledge-based approaches, suchas passwords and personal identification numbers (“PINs”), or ontoken-based approaches, such as swipe cards and photo identification, toestablish the identity of an individual. Despite being widely used,these are not very secure forms of identification. For example, it isestimated that hundreds of millions of dollars are lost annually incredit card fraud in the United States due to consumermisidentification.

Biometrics offers a reliable alternative for identifying an individual.Biometrics is the method of identifying an individual based on his orher physiological and behavioral characteristics. Common biometricmodalities include fingerprint, face recognition, hand geometry, voice,iris and signature verification, The Federal government will be aleading consumer of biometric applications deployed primarily forimmigration, airport, border, and homeland security. Wide scaledeployments of biometric applications such as the US-VISIT program arealready being done in the United States and else where in the world.

Despite advances in biometric identification systems, several obstacleshave hindered their deployment. Every biometric modality has some userswho have illegible biometrics. For example a recent NIST (NationalInstitute of Standards and Technology) study indicates that nearly 2 to5% of the population does not have legible fingerprints. Such userswould be rejected by a biometric fingerprint identification systemduring enrollment and verification. Handling such exceptions is timeconsuming and costly, especially in high volume scenarios such as anairport. Using multiple biometrics to authenticate an individual mayalleviate this problem.

Furthermore, unlike password or PIN based systems, biometric systemsinherently yield probabilistic results and are therefore not fullyaccurate. In effect a certain percentage of the genuine users will berejected (false non-match) and a certain percentage of impostors will beaccepted (false match) by existing biometric systems. High securityapplications require very low probability of false matches. For example,while authenticating immigrants and international passengers atairports, even a few false acceptances can pose a severe breach ofnational security. On the other hand false non matches lead to userinconvenience and congestion.

Existing systems achieve low false acceptance probabilities (also knownas False Acceptance Rate or “FAR”) only at the expense of higher falsenon-matching probabilities (also known as False-Rejection-Rate or“FRR”). It has been shown that multiple modalities can reduce FAR andFRR simultaneously. Furthermore, threats to biometric systems such asreplay attacks, spoofing and other subversive methods are difficult toachieve simultaneously for multiple biometrics, thereby makingmultimodal biometric systems more secure than single modal biometricsystems.

Systematic research in the area of combining biometric modalities isnascent and sparse. Over the years there have been many attempts atcombining modalities and many methods have been investigated, including“Logical And”, “Logical Or”, “Product Rule”, “Sum Rule”, “Max Rule”,“Min Rule”, “Median Rule”, “Majority Vote”, “Bayes' Decision”, and“Neyman-Pearson Test”. None of these methods has proved to provide lowFAR and FRR that is needed for modern security applications.

The need to address the challenges posed by applications using largebiometric databases is urgent. The US-VISIT program uses biometricsystems to enforce border and homeland security. Governments around theworld are adopting biometric authentication to implement Nationalidentification and voter registration systems. The Federal Bureau ofInvestigation maintains national criminal and civilian biometricdatabases for law enforcement.

Although large-scale databases are increasingly being used, the researchcommunity's focus is on the accuracy of small databases, whileneglecting the scalability and speed issues important to large databaseapplications. Each of the example applications mentioned above requiredatabases with a potential size in the tens of millions of biometricrecords. In such applications, response time, search and retrievalefficiency also become important in addition to accuracy.

As noted above, the invention may be used to determine whether a set ofbiometrics is acceptable for making a decision about whether atransaction should be authorized. The data set may be comprised ofinformation pieces about objects, such as people. Each object may haveat least two types of information pieces, that is to say the data setmay have at least two modalities. For example, each object representedin the database may by represented by two or more biometric samples, forexample, a fingerprint sample and an iris scan sample. A firstprobability partition array (“Pm(i,j)”) may be provided. The Pm(i,j) maybe comprised of probability values for information pieces in the dataset, each probability value in the Pm(i,j) corresponding to theprobability of an authentic match. Pm(i,j) may be similar to aNeyman-Pearson Lemma probability partition array. A second probabilitypartition array (“Pfm(i,j)”) may be provided, the Pm(i,j) beingcomprised of probability values for information pieces in the data set,each probability value in the Pfm(i,j) corresponding to the probabilityof a false match. Pfm(i,j) may be similar to a Neyman-Pearson Lemmaprobability partition array.

A method according to the invention may identify a no-match zone. Forexample, the no-match zone may be identified by identifying a firstindex set (“A”), the indices in set A being the (i,j) indices that havevalues in both Pfm(i,j) and Pm(i,j). A second index set (“Z∞”) may beidentified, the indices of Z∞ being the (i,j) indices in set A whereboth Pfm(i,j) is larger than zero and Pm(i,j) is equal to zero. FAR_(Z∞)may be determined, where FAR_(Z) _(∞) =1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j).FAR Z_(∞) may be compared to a desired false-acceptance-rate (“FAR”),and if FAR_(Z∞) is greater than the desired false-acceptance-rate, thanthe data set may be rejected for failing to provide an acceptablefalse-acceptance-rate. If FAR_(Z∞) is less than or equal to the desiredfalse-acceptance-rate, then the data set may be accepted, iffalse-rejection-rate is not important.

If false-rejection-rate is important, further steps may be executed todetermine whether the data set should be rejected. The method mayfurther include identifying a third index set ZM∞, the indices of ZM∞being the (i,j) indices in Z∞ plus those indices where both Pfm(i,j) andPm(i,j) are equal to zero. A fourth index set (“C”) may be identified,the indices of C being the (i,j) indices that are in A but not ZM∞. Theindices of C may be arranged such that

$\frac{{P_{fm}\left( {i,j} \right)}_{k}}{{P_{m}\left( {i,j} \right)}_{k}}>=\frac{{P_{fm}\left( {i,j} \right)}_{k + 1}}{{P_{m}\left( {i,j} \right)}_{k + 1}}$to provide an arranged C index. A fifth index set (“Cn”) may beidentified. The indices of Cn may be the first N (i,j) indices of thearranged C index, where N is a number for which the following is true:FAR_(Z) _(∞) _(∪C) _(N) =1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j)−Σ_((i,j)εC) _(N)P_(fm)(i,j)≦FAR. The FRR may be determined, where FRR=Σ_((i,j)εC) _(N)P_(m)(i,j), and compared to a desired false-rejection-rate. If FRR isgreater than the desired false-rejection-rate, then the data set may berejected, even though FAR_(Z∞) is less than or equal to the desiredfalse-acceptance-rate. Otherwise, the data set may be accepted.

In another method according to the invention, the false-rejection-ratecalculations and comparisons may be executed before thefalse-acceptance-rate calculations and comparisons. In such a method, afirst index set (“A”) may be identified, the indices in A being the(i,j) indices that have values in both Pfm(i,j) and Pm(i,j). A secondindex set (“Z∞”) may be identified, the indices of Z∞ being the (i,j)indices of A where Pm(i,j) is equal to zero. A third index set (“C”) maybe identified, the indices of C being the (i,j) indices that are in Abut not Z∞. The indices of C may be arranged such that

$\frac{{P_{fm}\left( {i,j} \right)}_{k}}{{P_{m}\left( {i,j} \right)}_{k}}>=\frac{{P_{fm}\left( {i,j} \right)}_{k + 1}}{{P_{m}\left( {i,j} \right)}_{k + 1}}$to provide an arranged C index, and a fourth index set (“Cn”) may beidentified. The indices of Cn may be the first N (i,j) indices of thearranged C index, where N is a number for which the following is true:FAR_(Z) _(∞) _(∪C) _(N) =1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j)−Σ_((i,j)εC) _(N)P_(fm)(i,j)≦FAR. The ERR may be determined, where FRR=Σ_((i,j)εC) _(N)P_(m)(i,j), and compared to a desired false-rejection-rate. If the FRRis greater than the desired false-rejection-rate, then the data set maybe rejected. If the FRR is less than or equal to the desiredfalse-rejection-rate, then the data set may be accepted, iffalse-acceptance-rate is not important. If false-acceptance-rate isimportant, then the FAR_(Z∞), may be determined, where FAR_(Z) _(∞)=1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j). The FAR_(Z∞) may be compared to adesired false-acceptance-rate, and if FAR_(Z∞) is greater than thedesired false-acceptance-rate, then the data set may be rejected eventhough FRR is less than or equal to the desired false-rejection-rate.Otherwise, the data set may be accepted.

The invention may also be embodied as a computer readable memory devicefor executing any of the methods described above.

Preliminary Considerations: To facilitate discussion of the invention itmay be beneficial to establish some terminology and a mathematicalframework. Biometric fusion may be viewed as an endeavor in statisticaldecision theory [1] [2]; namely, the testing of simple hypotheses. Thisdisclosure uses the term simple hypothesis as used in standardstatistical theory: the parameter of the hypothesis is stated exactly.The hypotheses that a biometric score “is authentic” or “is from animpostor” are both simple hypotheses.

In a match attempt, the N-scores reported from N<∞ biometrics is calledthe test observation. The test observation, xεS, where S⊂R^(N)(Euclidean N-space) is the score sample space, is a vector function of arandom variable X from which is observed a random sample(X₁,X₂, . . . ,X_(N)). The distribution of scores is provided by the jointclass-conditional probability density function (“pdf”):ƒ(x|θ)=ƒ(x ₁ , x ₂ , . . . , x _(N)|θ)  (1)where θ equals either Au to denote the distribution of authentic scoresor Im to denote the distribution of impostor scores. So, if θ=Au,Equation (1) is the pdf for authentic distribution of scores and if θ=Imit is the pdf for the impostor distribution. It is assumed thatƒ(x|θ)=ƒ(x₁,x₂, . . . , x_(N)|θ)>0 on S.

Given a test observation, the following two simple statisticalhypotheses are observed: The null hypothesis, H₀, states that a testobservation is an impostor; the alternate hypothesis, H₁, states that atest observation is an authentic match. Because there are only twochoices, the fusion logic will partition S into two disjoint sets:R_(Au)⊂S and R_(Im)⊂S, where R_(Au)∩R_(Im)=Ø and R_(Au)∪R_(Im)=S.Denoting the compliment of a set A by A^(C), so that R_(Au) ^(C)=R_(Im)and R_(Im) ^(C)=R_(Au).

The decision logic is to accept H₁ (declare a match) if a testobservation, x, belongs to R_(Au) or to accept H₀ (declare no match andhence reject H₁) if x belongs to R_(Im). Each hypothesis is associatedwith an error type: A Type I error occurs when H₀ is rejected (acceptH₁) when H₀ is true; this is a false accept (FA); and a Type II erroroccurs when H₁ is rejected (accept H₀) when H₁ is true; this is a falsereject (FR). Thus denoted, respectively, the probability of making aType I or a Type II error follows:

$\begin{matrix}{{FAR} = {P_{{Type}\mspace{11mu} I} = {{P\left( {x \in R_{Au}} \middle| {Im} \right)} = {\int_{R_{Au}}{{f\left( x \middle| {Im} \right)}\ {\mathbb{d}x}}}}}} & (2) \\{{FRR} = {P_{{Type}\mspace{11mu}{II}} = {{P\left( {x \in R_{Im}} \middle| {Au} \right)} = {\int_{R_{Im}}{{f\left( x \middle| {Au} \right)}\ {\mathbb{d}x}}}}}} & (3)\end{matrix}$So, to compute the false-acceptance-rate (FAR), the impostor score's pdfis integrated over the region which would declare a match (Equation 2).The false-rejection-rate (FRR) is computed by integrating the pdf forthe authentic scores over the region in which an impostor (Equation 3)is declared. The correct-acceptance-rate (CAR) is 1−FRR.

The class-conditional pdf for each individual biometric, the marginalpdf, is assumed to have finite support; that is, the match scoresproduced by the i^(th) biometric belong to a closed interval of the realline, γ_(i)=[α_(i), ω_(i)], where −∞<α_(i)<ω_(i)<+∞. Two observationsare made. First, the sample space, S, is the Cartesian product of theseintervals, i.e., S=γ₁×γ₂× . . . ×γ_(N). Secondly, the marginal pdf,which we will often reference, for any of the individual biometrics canbe written in terns of the joint pdf:

$\begin{matrix}{{{f_{i}\left( x_{i} \middle| \theta \right)} = {\int_{\gamma_{j}\forall}^{\;}{\int_{j \neq i}^{\;}{\ldots{\int{{f\left( x \middle| \theta \right)}{\mathbb{d}x}}}}}}}\ } & (4)\end{matrix}$For the purposes of simplicity the following definitions are stated:

-   -   Definition 1: In a test of simple hypothesis, the        correct-acceptance-rate, CAR=1−FRR is known as the power of the        test.    -   Definition 2: For a fixed FAR, the test of simple H₀ versus        simple H₁ that has the largest CAR is called most powerful.    -   Definition 3: Verification is a one-to-one template comparison        to authenticate a person's identity (the person claims their        identity).    -   Definition 4: Identification is a one-to-many template        comparison to recognize an individual (identification attempts        to establish a person's identity without the person having to        claim an identity),        The receiver operation characteristics (ROC) curve is a plot of        CAR versus FAR, an example of which is shown in FIG. 8. Because        the ROC shows performance for all possible specifications of        FAR, as can be seen in the FIG. 8, it is an excellent tool for        comparing performance of competing systems, and we use it        throughout our analyses.

Fusion and Decision Methods: Because different applications havedifferent requirements for error rates, there is an interest in having afusion scheme that has the flexibility to allow for the specification ofa Type-I error yet have a theoretical basis for providing the mostpowerful test, as defined in Definition 2. Furthermore, it would bebeneficial if the fusion scheme could handle the general problem, sothere are no restrictions on the underlying statistics. That is to saythat:

-   -   Biometrics may or may not be statistically independent of each        other,    -   The class conditional pdf can be multi-modal (have local        maximums) or have no modes.    -   The underlying pdf is assumed to be nonparametric (no set of        parameters define its shape, such as the mean and standard        deviation for a Gaussian distribution).

A number of combination strategies were examined, all of which arelisted by Jain [4] on page 243. Most of the schemes, such as“Demptster-Shafer” and “fuzzy integrals” involve training. These wererejected mostly because of the difficulty in analyzing and controllingtheir performance mechanisms in order to obtain optimal performance.Additionally, there was an interest in not basing the fission logic onempirical results. Strategies such as SUM, MEAN, MEDIAN, PRODUCT, MIN,MAX were likewise rejected because they assume independence betweenbiometric features.

When combining two biometrics using a Boolean “AND” or “OR” it is easilyshown to be suboptimal when the decision to accept or reject H₁ is basedon fixed score thresholds. However, the “AND” and the “OR” are the basicbuilding blocks of verification (OR) and identification (AND). Since weare focused on fixed score thresholds, making an optimal decision toaccept or reject H₁ depends on the structure of: R_(Au)⊂S and R_(Im)⊂S,and simple thresholds almost always form suboptimal partitions.

If one accepts that accuracy dominates the decision making process andcost dominates the combination strategy, certain conclusions may bedrawn. Consider a two-biometric verification system for which a fixedFAR has been specified. In that system, a person's identity isauthenticated if H₁ is accepted by the first biometric OR if accepted bythe second biometric. If H₁ is rejected by the first biometric AND thesecond biometric, then manual intervention is required. If a cost isassociated with each stage of the verification process, a cost functioncan be formulated. It is a reasonable assumption to assume that thefewer people that filter down from the first biometric sensor to thesecond to the manual check, the cheaper the system.

Suppose a fixed threshold is used as the decision making process toaccept or reject H₁ at each sensor and solve for thresholds thatminimize the cost function. It will obtain settings that minimize cost,but it will not necessarily be the cheapest cost. Given a more accurateway to make decisions, the cost will drop. And if the decision methodguarantees the most powerful test at each decision point it will havethe optimal cost.

It is clear to us that the decision making process is crucial. A surveyof methods based on statistical decision theory reveals many powerfultests such as the Maximum-Likelihood Test and Bayes' Test. Each requiresthe class conditional probability density function, as given byEquation 1. Some, such as the Bayes' Test, also require the a prioriprobabilities P(H₀) and P(H₁), which are the frequencies at which wewould expect an impostor and authentic match attempt. Generally, thesetests do not allow the flexibility of specifying a FAR—they minimizemaking a classification error.

In their seminal paper of 1933 [3], Neyman and Pearson presented a lemmathat guarantees the most powerful test for a fixed FAR requiring onlythe joint class conditional pdf for H₀ and H₁. This test may be used asthe centerpiece of the biometric fusion logic employed in the invention.The Neyman-Pearson Lemma guarantees the validity of the test. The proofof the Lemma is slightly different than those found in other sources,but the reason for presenting it is because it is immediately amenableto proving the Corollary to the Neyman Pearson Lemma. The corollarystates that fusing two biometric scores with Neyman-Pearson, alwaysprovides a more powerful test than either of the component biometrics bythemselves. The corollary is extended to state that fusing N biometricscores is better than fusing N−1 scores

Deriving the Neyman-Pearson Test: Let FAR=α be fixed. It is desired tofind R_(Au)⊂S such that

α = ∫_(R_(Au))f(x|Im) 𝕕x  and  ∫_(R_(Au))f(x|Au) 𝕕xis most powerful. To do this the objective set function is formed thatis analogous to Lagrange's Method:

u(R_(Au)) = ∫_(R_(Au))f(x|Au) 𝕕x − λ[∫_(R_(Au))f(x|Im) 𝕕x − α]where λ≧0 is an undetermined Lagrange multiplier and the external valueof u is subject to the constraint

α = ∫_(R_(Au))f(x|Im) 𝕕x.Rewriting the above equation as

u(R_(Au)) = ∫_(R_(Au))[f(x|Au)  − f(x|Im)]𝕕x + λαwhich ensures that the integrand in the above equation is positive forall xεR_(Au). Let λ≧0, and, recalling that the class conditional pdf ispositive on S is assumed, define

$R_{Au} = {\left\{ {{x{\text{:}\left\lbrack {{f\left( x \middle| {Au} \right)}\  - {\lambda\;{f\left( x \middle| {Im} \right)}}} \right\rbrack}} > 0} \right\} = \left\{ {{x\text{:}\frac{f\left( x \middle| {Au} \right)}{f\left( x \middle| {Im} \right)}} > \lambda} \right\}}$Then u is a maximum if λ is chosen such that

α = ∫_(R_(Au))f(x|Im) 𝕕xis satisfied.

The Neyman-Pearson Lemma: Proofs of the Neyman-Pearson Lemma can befound in their paper [4] or in many texts [1] [2]. The proof presentedhere is somewhat different. An “in-common” region, R_(IC) is establishedbetween two partitions that have the same FAR. It is possible thisregion is empty. Having R_(IC) makes it easier to prove the corollarypresented.

Neyman-Pearson Lemma: Given the joint class conditional probabilitydensity functions for a system of order N in making a decision with aspecified FAR=α, let λ be a positive real number, and let

$\begin{matrix}{R_{Au} = \left\{ {x \in S} \middle| {\frac{f\left( x \middle| {Au} \right)}{f\left( x \middle| {Im} \right)} > \lambda} \right\}} & (5) \\{R_{Au}^{C} = {R_{Im} = \left\{ {x \in S} \middle| {\frac{f\left( x \middle| {Au} \right)}{f\left( x \middle| {Im} \right)} \leq \lambda} \right\}}} & (6)\end{matrix}$such that

$\begin{matrix}{\alpha = {\int_{R_{Au}}{{f\left( x \middle| {Im} \right)}\ {{\mathbb{d}x}.}}}} & (7)\end{matrix}$then R_(Au) is the best critical region for declaring a match—it givesthe largest correct-acceptance-rate (CAR), hence R_(λ) ^(C) gives thesmallest FRR.

Proof: The lemma is trivially true if R_(Au) is the only region forwhich Equation (7) holds. Suppose R_(φ)≠R_(Au) with m(R_(φ)∩R_(Au)^(C))>0 (this excludes sets that are the same as R_(Au) except on a setof measure zero that contribute nothing to the integration), is anyother region such that

$\begin{matrix}{\alpha = {\int_{R_{\phi}}{{f\left( x \middle| {Im} \right)}\ {{\mathbb{d}x}.}}}} & (8)\end{matrix}$Let R_(IC)=R_(φ)∩R_(Au), which is the “in common” region of the two setsand may be empty. The following is observed:

$\begin{matrix}{{\int_{R_{Au}}{{f\left( x \middle| \theta \right)}\ {\mathbb{d}x}}} = {{\int_{R_{Au} - R_{1C}}{{f\left( x \middle| \theta \right)}\ {\mathbb{d}x}}} + {\int_{R_{1C}}{{f\left( x \middle| \theta \right)}\ {\mathbb{d}x}}}}} & (9) \\{{\int_{R_{\phi}}{{f\left( x \middle| \theta \right)}\ {\mathbb{d}x}}} = {{\int_{R_{\phi} - R_{1C}}{{f\left( x \middle| \theta \right)}\ {\mathbb{d}x}}} + {\int_{R_{1C}}{{f\left( x \middle| \theta \right)}\ {\mathbb{d}x}}}}} & (10)\end{matrix}$If R_(Au) has a better CAR than R_(φ) then it is sufficient to provethat

$\begin{matrix}{{{\int_{R_{Au}}{{f\left( x \middle| {Au} \right)}\ {\mathbb{d}x}}} - {\int_{R_{\phi}}{{f\left( x \middle| {Au} \right)}\ {\mathbb{d}x}}}} > 0} & (11)\end{matrix}$From Equations (7), (8), (9), and (10) it is seen that (11) holds if

$\begin{matrix}{{{\int_{R_{Au} - R_{IC}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}} - {\int_{R_{\phi} - R_{IC}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}}} > 0} & (12)\end{matrix}$Equations (7), (8), (9), and (10) also gives

$\begin{matrix}{\alpha_{R_{\phi} - R_{IC}} = {{\int_{R_{Au} - R_{IC}}^{\;}{{f\left( x \middle| {Im} \right)}{\mathbb{d}x}}} = {\int_{R_{\phi} - R_{IC}}^{\;}{{f\left( x \middle| {Im} \right)}{\mathbb{d}x}}}}} & (13)\end{matrix}$When xεR_(Au)−R_(IC)⊂R_(Au) it is observed from (5) that

$\begin{matrix}{{{\int_{R_{Au} - R_{IC}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}} > {\int_{R_{Au} - R_{IC}}^{\;}{\lambda\;{f\left( x \middle| {Im} \right)}{\mathbb{d}x}}}} = {\lambda\;\alpha_{R_{\phi} - R_{IC}}}} & (14)\end{matrix}$and when xεR_(φ)−R_(IC)⊂R_(Au) ^(C) it is observed from (6) that

$\begin{matrix}{{{\int_{R_{\phi} - R_{IC}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}} \leq {\int_{R_{\phi} - R_{IC}}^{\;}{\lambda\;{f\left( x \middle| {Im} \right)}{\mathbb{d}x}}}} = {\lambda\;\alpha_{R_{\phi} - R_{IC}}}} & (15)\end{matrix}$Equations (14) and (15) give

$\begin{matrix}{{\int_{R_{Au} - R_{IC}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}} > {\lambda\;\alpha_{R_{\phi} - R_{IC}}} \geq {\int_{R_{\phi} - R_{IC}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}}} & (16)\end{matrix}$This establishes (12) and hence (11), which proves the lemma. // In thisdisclosure, the end of a proof is indicated by double slashes “//”.

Corollary: Neyman-Pearson Test Accuracy Improves with AdditionalBiometrics; The fact that accuracy improves with additional biometricsis an extremely important result of the Neyman-Pearson Lemma. Under theassumed class conditional densities for each biometric, theNeyman-Pearson Test provides the most powerful test over any other testthat considers less than all the biometrics available. Even if abiometric has relatively poor performance and is highly correlated toone or more of the other biometrics, the fused CAR is optimal. Thecorollary for N-biometrics versus a single component biometric follows.

Corollary to the Neyman Pearson Lemma: Given the joint class conditionalprobability density functions for an N-biometric system, choose α=FARand use the Neyman-Pearson Test to find the critical region R_(Au) thatgives the most powerful test for the N-biometric system

$\begin{matrix}{{CAR}_{R_{Au}} = {\int_{R_{Au}}^{\;}{{f\left( x \middle| {Au} \right)}{{\mathbb{d}x}.}}}} & (17)\end{matrix}$Consider the i^(th) biometric. For the same α=FAR, use theNeyman-Pearson Test to find the critical collection of disjointintervals I_(Au) ⊂R¹ that gives the most powerful test for the singlebiometric, that is,

$\begin{matrix}{{CAR}_{I_{Au}} = {\int_{I\;{Au}}^{\;}{{f\left( x_{i} \middle| {Au} \right)}{\mathbb{d}x}}}} & (18) \\{then} & \; \\{{CAR}_{R_{Au}} \geq {{CAR}_{I_{Au}}.}} & (19)\end{matrix}$Proof: Let R_(i)=I_(Au)×γ₁×γ₂× . . . ×γ_(N)⊂R^(N), where the Cartesianproducts are taken over all the γ_(k) except for k=i. From (4) themarginal pdf can be recast in terms of the joint pdf

$\begin{matrix}{{\int_{R_{i}}^{\;}{{f\left( x \middle| \theta \right)}{\mathbb{d}x}}} = {\int_{I_{Au}}^{\;}{{f\left( x_{i} \middle| \theta \right)}{\mathbb{d}x}}}} & (20)\end{matrix}$First, it will be shown that equality holds in (19) if and only ifR_(Au)=R_(i). Given that R_(Au)=R_(i) except on a set of measure zero,i.e., m(R_(Au) ^(C)∩R_(i))=0, then clearly CAR_(R) _(Au) =CAR_(I) _(Au). On the other hand, assume CAR_(R) _(Au) =CAR_(I) _(Au) and m(R_(Au)^(C)∩R_(i))>0, that is, the two sets are measurably different. But thisis exactly the same condition previously set forth in the proof of theNeyman-Pearson Lemma from which from which it was concluded that CAR_(R)_(Au) >CAR_(I) _(Au) , which is a contradiction. Hence equality holds ifand only if R_(Au)=R_(i). Examining the inequality situation in equation(19), given that R_(Au)≠R_(i) such that m(R_(Au) ^(C)∩R_(i))>0, then itis shown again that the exact conditions as in the proof of theNeyman-Pearson Lemma have been obtained from which we conclude thatCAR_(R) _(Au) >CAR_(I) _(Au) , which proves the corollary.

Examples have been built such that CAR_(R) _(Au) =CAR_(I) _(Au) but itis hard to do and it is unlikely that such densities would be seen in areal world application. Thus, it is safe to assume that CAR_(R) _(Au)>CAR_(I) _(Au) is almost always true.

The corollary can be extended to the general case. The fusion of Nbiometrics using Neyman-Pearson theory always results in a test that isas powerful as or more powerful than a test that uses any combination ofM<N biometrics. Without any loss to generality, arrange the labels sothat the first M biometrics are the ones used in the M<N fusion. Chooseα=FAR and use the Neyman-Pearson Test to find the critical regionR_(M)⊂R^(M) that gives the most powerful test for the M-biometricsystem. Let R_(N)=R_(M)×γ_(M+1)×γ_(M+2)× . . . ×γ_(N)⊂R^(N), where theCartesian products are taken over all the γ-intervals not used in the Mbiometrics combination. Then writing

$\begin{matrix}{\int_{R_{N}}^{\;}{{f\left( x \middle| \theta \right)}{\mathbb{d}x}{\int_{R_{M}}^{\;}{{f\left( {x_{1},{x_{2}\mspace{11mu}\ldots}\mspace{11mu},\left. x_{M} \middle| \theta \right.} \right)}{\mathbb{d}x}}}}} & (21)\end{matrix}$gives the same construction as in (20) and the proof flows as it did forthe corollary.

We now state and prove the following five mathematical propositions. Thefirst four propositions are necessary to proving the fifth proposition,which will be cited in the section that details the invention. For eachof the propositions we will use the following: Let r={r₁,r₂, . . . ,r_(n) } be a sequence of real valued ratios such that r₁≧r₂≧ . . .≧r_(n). For each r_(i) we know the numerator and denominator of theratio, so that

${r_{i} = \frac{n_{i}}{d_{i}}},$n_(i), d_(i)>0∀i.

Proposition #1: For r_(i), r_(i+1)εr and

${r_{i} = \frac{n_{i}}{d_{i}}},{r_{i + 1} = \frac{n_{i + 1}}{d_{i + 1}}},$then

${\frac{n_{i} + n_{i + 1}}{d_{i} + d_{i + 1}} \leq \frac{n_{i}}{d_{i}}} = {r_{i}.}$Proof: Because r_(i), r_(i+1)εr we have

$\left. {\frac{n_{i + 1}}{d_{i + 1}} \leq \frac{n_{i}}{d_{i}}}\Rightarrow{\frac{d_{i}n_{i + 1}}{n_{i}d_{i + 1}} \leq 1}\Rightarrow{{\frac{d_{i}}{n_{i}}\left( {n_{i} + n_{i + 1}} \right)} \leq {d_{i} + d_{i + 1}}}\Rightarrow{\frac{n_{i} + n_{i + 1}}{d_{i} + d_{i + 1}} \leq \frac{n_{i}}{d_{i}}} \right. = r_{i}$

Proposition #2: For r_(i), r_(i+1)εr and

${r_{i} = \frac{n_{i}}{d_{i}}},{r_{i + 1} = \frac{n_{i + 1}}{d_{i + 1}}},$then

$r_{i + 1} = {\frac{n_{i + 1}}{d_{i + 1}} \leq {\frac{n_{i} + n_{i + 1}}{d_{i} + d_{i + 1}}.}}$Proof: The proof is the same as for proposition #1 except the order isreversed.

Proposition #3: For

${r_{i} = \frac{n_{i}}{d_{i}}},$i=(1, . . . , n), then

${\frac{\sum\limits_{i = m}^{M}n_{i}}{\sum\limits_{i = m}^{M}d_{i}} \leq \frac{n_{m}}{d_{m}}},$1≦m≦M≦n. Proof: The proof is by induction. We know from proposition #1that the result holds for N=1, that is

$\frac{n_{i} + n_{i + 1}}{d_{i} + d_{i + 1}} \leq {\frac{n_{i}}{d_{i}}.}$Now assume it holds for any N<1, we need to show that it holds for N+1.Let m=2 and M =N+1 (note that this is the N^(th) case and not the N+1case), then we assume

$\left. {\frac{\sum\limits_{i = 2}^{N + 1}n_{i}}{\sum\limits_{i = 2}^{N + 1}d_{i}} \leq \frac{n_{2}}{d_{2}} \leq \frac{n_{1}}{d_{1}}}\Rightarrow{{\frac{d_{1}}{n_{1}}\frac{\sum\limits_{i = 2}^{N + 1}n_{i}}{\sum\limits_{i = 2}^{N + 1}d_{i}}} \leq 1}\Rightarrow{{\frac{d_{1}}{n_{1}}{\sum\limits_{i = 2}^{N + 1}n_{i}}} \leq {\sum\limits_{i = 2}^{N + 1}d_{i}}}\Rightarrow{{{\frac{d_{1}}{n_{1}}{\sum\limits_{i = 2}^{N + 1}n_{i}}} + d_{1}} \leq {{\sum\limits_{i = 2}^{N + 1}d_{i}} + d_{1}}} \right. = {\left. {\sum\limits_{i = 1}^{N + 1}d_{i}}\Rightarrow{{d_{1}\left( {1 + \frac{\sum\limits_{i = 2}^{N + 1}n_{i}}{n_{1}}} \right)} \leq {\sum\limits_{i = 1}^{N + 1}d_{i}}}\Rightarrow{d_{1}\left( \frac{n_{1} + {\sum\limits_{i = 2}^{N + 1}n_{i}}}{n_{1}} \right)} \right. = \left. {{\frac{d_{1}}{n_{1}}{\sum\limits_{i = 1}^{N + 1}n_{i}}} \leq {\sum\limits_{i = 1}^{N + 1}d_{i}}}\Rightarrow{\frac{\sum\limits_{i = 1}^{N + 1}n_{i}}{\sum\limits_{i = 1}^{N + 1}d_{i}} \leq \frac{n_{1}}{d_{1}}} \right.}$which is the N+1 case as was to be shown.

Proposition #4: For

${r_{i} = \frac{n_{i}}{d_{i}}},$i=(1, . . . , n), then

${\frac{n_{M}}{d_{M}} \leq \frac{\sum\limits_{i = m}^{M}n_{i}}{\sum\limits_{i = m}^{M}d_{i}}},$1≦m≦M≦n.Proof: As with proposition #3, the proof is by induction. We know fromproposition #2 that the result holds for N=1, that is

$\frac{n_{i + 1}}{d_{i + 1}} \leq {\frac{n_{i} + n_{i + 1}}{d_{i} + d_{i + 1}}.}$The rest of the proof follows the same format as for proposition #3 withthe order reversed.

Proposition #5: Recall that the r is an ordered sequence of decreasingratios with known numerators and denominators. We sum the first Nnumerators to get S_(n) and sum the first N denominators to get S_(d).We will show that for the value S_(n), there is no other collection ofratios in r that gives the same S_(n) and a smaller S_(d). For

${r_{i} = \frac{n_{i}}{d_{i}}},$i=(1, . . . , n), let S be the sequence of the first N terms of r, withthe sum of numerators given by S_(n)=Σ_(i=1) ^(N)n_(i), and the sum ofdenominators by S_(d)=Σ_(i=1) ^(N)d_(i), 1≦N≦n. Let S′ be any othersequence of ratios in r, with numerator sum S′_(n)=Σn_(i) anddenominator sum S′_(d)=Σd_(i) such that S′_(n)=S_(n), then we haveS_(d)≦S′_(d). Proof: The proof is by contradiction. Suppose theproposition is not true, that is, assume another sequence, S′, existssuch that S′_(d)<S_(d). For this sequence, define the indexing setsA={indices that can be between 1 and N inclusive} and B={indices thatcan be between N+1 and n inclusive}. We also define the indexing setA^(c)={indices between 1 and N inclusive and not in A}, which meansA∩A^(c)=Ø and A∪A^(c)={all indices between 1 and N inclusive}. Then ourassumption states:

$S_{n}^{\prime} = {{{\sum\limits_{i \in A}\; n_{i}} + {\sum\limits_{i \in B}\; n_{i}}} = S_{n}}$and$S_{d}^{\prime} = {{{\sum\limits_{i \in A}\; d_{i}} + {\sum\limits_{i \in B}^{\;}\; d_{i}}} < {S_{d}.}}$This implies that

$\begin{matrix}{{{{\sum\limits_{i \in A}\; d_{i}} + {\sum\limits_{i \in B}^{\;}\; d_{i}}} < {\sum\limits_{i = 1}^{N}\; d_{i}}} = \left. {{\sum\limits_{i \in A}\; d_{i}} + {\sum\limits_{i \in A^{c}}^{\;}\;{d_{i}.}}}\Rightarrow{{\sum\limits_{i \in B}^{\;}\; d_{i}} < {\sum\limits_{i \in A^{c}}^{\;}\; d_{i}}}\Rightarrow{\frac{1}{\sum\limits_{i \in A^{c}}^{\;}\; d_{i}} < \frac{1}{\sum\limits_{i \in B}^{\;}\; d_{i}}} \right.} & (22)\end{matrix}$Because the numerators of both sequences are equal, we can write

$\begin{matrix}{{{\sum\limits_{i \in A}\; n_{i}} + {\sum\limits_{i \in B}\; n_{i}}} = {\left. {\sum\limits_{i \in {A\bigcup A^{c}}}^{\;}\; n_{i}}\Rightarrow{\sum\limits_{i \in B}\; n_{i}} \right. = {{{\sum\limits_{i \in {A\bigcup A^{c}}}^{\;}\; n_{i}} - {\sum\limits_{i \in A}\; n_{i}}} = {\sum\limits_{i \in A^{c}}\; n_{i}}}}} & (23)\end{matrix}$Combining (22) and (23), and from propositions #3 and #4, we have

$\begin{matrix}{r_{N} = {\frac{n_{N}}{d_{N}} \leq \frac{\sum\limits_{i \in A^{c}}^{\;}\; n_{i}}{\sum\limits_{i \in A^{c}}^{\;}\; d_{i}} < \frac{\sum\limits_{i \in B^{\;}}^{\;}\; n_{i}}{\sum\limits_{i \in B^{\;}}^{\;}\; d_{i}} \leq \frac{n_{N + 1}}{d_{N + 1}}}} \\{= r_{{N + 1},}}\end{matrix}$which contradicts the fact that r_(N)≧r_(N+1), hence the validity of theproposition follows.

Cost Functions for Optimal Verification and Identification: In thediscussion above, it is assumed that all N biometric scores aresimultaneously available for fusion. The Neyman-Pearson Lemma guaranteesthat this provides the most powerful test for a fixed FAR. In practice,however, this could be an expensive way of doing business. If it isassumed that all aspects of using a biometric system, time, risk, etc.,can be equated to a cost, then a cost function can be constructed. Theinventive process described below constructs the cost function and showshow it can be minimized using the Neyman-Pearson test. Hence, theNeyman-Pearson theory is not limited to “all-at-once” fusion; it can beused for serial, parallel, and hierarchal systems.

Having laid a basis for the invention, a description of two costfunctions is now provided as a mechanism for illustrating an embodimentof the invention. A first cost function for a verification system and asecond cost function for an identification system are described. Foreach system, an algorithm is presented that uses the Neyman-Pearson testto minimize the cost function for a second order biometric system, thatis a biometric system that has two modalities. The cost functions arepresented for the general case of N-biometrics. Because minimizing thecost function is recursive, the computational load grows exponentiallywith added dimensions. Hence, an efficient algorithm is needed to handlethe general case.

First Cost Function—Verification System. A cost function for a 2-stagebiometric verification (one-to-one) system will be described, and thenan algorithm for minimizing the cost function will be provided. In a2-stage verification system, a subject may attempt to be verified by afirst biometric device. If the subject's identity cannot beauthenticated, the subject may attempt to be authenticated by a secondbiometric. If that fails, the subject may resort to manualauthentication. For example, manual authentication may be carried out byinterviewing the subject and determining their identity from othermeans.

The cost for attempting to be authenticated using a first biometric, asecond biometric, and a manual check are c₁, c₂, and c₃, respectively.The specified FAR_(sys) is a system false-acceptance-rate, i.e. the rateof falsely authenticating an impostor includes the case of it happeningat the first biometric or the second biometric. This implies that thefirst-biometric station test cannot have a false-acceptance-rate, FAR₁,that exceeds FAR_(sys). Given a test with a specified FAR₁, there is anassociated false-rejection-rate, FRR₁, which is the fraction of subjectsthat, on average, are required to move on to the second biometricstation. The FAR required at the second station is FAR₂=FAR_(sys)−FAR₁.It is known that P(A∪B)=P(A)+P(B)−P(A)P(B), so the computation of FAR₂appears imperfect. However, if FAR_(sys)=P(A∪B) and FAR₁=P(A), in theconstruction of the decision space, it is intended thatFAR₂=P(B)−P(A)P(B).

Note that FAR₂ is a function of the specified FAR_(sys) and the freelychosen FAR₁; FAR₂ is not a free parameter. Given a biometric test withthe computed FAR₂, there is an associated false-rejection-rate, FRR₂ bythe second biometric test, which is the fraction of subjects that arerequired to move on to a manual check. This is all captured by thefollowing cost function:Cost=c ₁+FRR₁(c ₂ +c ₃FRR₂)  (30)

There is a cost for every choice of FAR₁≦FAR_(sys), so the Cost inEquation 30 is a function of FAR₁. For a given test method, there existsa value of FAR₁ that yields the smallest cost and we present analgorithm to find that value. In a novel approach to the minimization ofEquation 30, a modified version of the Neyman-Pearson decision test hasbeen developed so that the smallest cost is optimally small.

An algorithm is outlined below. The algorithm seeks to optimallyminimize Equation 30. To do so, we (a) set the initial cost estimate toinfinity and, (b) for a specified FAR_(sys), loop over all possiblevalues of FAR₁≦FAR_(sys). In practice, the algorithm may use a uniformlyspaced finite sample of the infinite possible values. The algorithm mayproceed as follows: (c) set FAR₁=FAR_(sys), (d) set Cost=∞, and (e) loopover possible FAR₁ values. For the first biometric at the current FAR₁value, the algorithm may proceed to (f) find the optimal Match-Zzone,R₁, and (g) compute the correct-acceptance-rate over R₁ by:

$\begin{matrix}{{CAR}_{1} = {\int_{R_{1}}^{\;}{{f\left( {x❘{Au}} \right)}\ {\mathbb{d}x}}}} & (31)\end{matrix}$and (h) determine FRR₁ using FRR₁=1−CAR₁.

Next, the algorithm may test against the second biometric. Note that theregion R₁ of the score space is no longer available since the firstbiometric test used it up. The Neyman-Pearson test may be applied to thereduced decision space, which is the compliment of R₁. So, at this time,(h) the algorithm may compute FAR₂=FAR_(sys)−FAR₁, and FAR₂ may be (i)used in the Neyman-Pearson test to determine the most powerful test,CAR₂, for the second biometric fused with the first biometric over thereduced decision space R₁ ^(C). The critical region for CAR₂ is R₂,which is disjoint from R₁ by our construction. Score pairs that resultin the failure to be authenticated at either biometric station must fallwithin the region R₃=(R₁∪R₂)^(C), from which it is shown that

${FRR}_{2} = \frac{\int_{R_{3}}^{\;}{{f\left( {x❘{Au}} \right)}\ {\mathbb{d}x}}}{{FRR}_{1}}$

The final steps in the algorithm are (j) to compute the cost usingEquation 30 at the current setting of FAR₁ using FRR₁and FRR₂, and (k)to reset the minimum cost if cheaper, and keep track of the FAR₁responsible for the minimum cost.

To illustrate the algorithm, an example is provided. Problems arisingfrom practical applications are not to be confused with the validity ofthe Neyman-Pearson theory. Jain states in [5]: “In case of a largernumber of classifiers and relatively small training data, a classifiermy actually degrade the performance when combined with other classifiers. . . ” This would seem to contradict the corollary and its extension.However, the addition of classifiers does not degrade performancebecause the underlying statistics are always true and the corollaryassumes the underlying statistics. Instead, degradation is a result ofinexact estimates of sampled densities. In practice, a user may beforced to construct the decision test from the estimates, and it iserrors in the estimates that cause a mismatch between predictedperformance and actual performance.

Given the true underlying class conditional pdf for, H₀ and H₁, thecorollary is true. This is demonstrated with a challenging example usingup to three biometric sensors. The marginal densities are assumed to beGaussian distributed. This allows a closed form expression for thedensities that easily incorporates correlation. The general n^(th) orderform is well known and is given by

$\begin{matrix}{{f\left( {x❘\theta} \right)} = {\frac{1}{\left( {2\pi} \right)^{\frac{n}{2}}{C}^{1/2}}{\exp\left( {- {\frac{1}{2}\left\lbrack {\left( {x - \mu} \right)^{T}{C^{- 1}\left( {x - \mu} \right)}} \right\rbrack}} \right)}}} & (32)\end{matrix}$where μ is the mean and C is the covariance matrix. The mean (μ) and thestandard deviation (σ) for the marginal densities are given in Table 1.Plots of the three impostor and three authentic densities are shown inError! Reference source not found.

TABLE 1 Biometric Impostors Authentics Number μ Σ μ σ #1 0.3 0.06 0.650.065 #2 0.32 0.065 0.58 0.07 #3 0.45 0.075 0.7 0.08

TABLE 2 Correlation Coefficient Impostors Authentics ρ₁₂ 0.03 0.83 ρ₁₃0.02 0.80 ρ₂₃ 0.025 0.82

To stress the system, the authentic distributions for the threebiometric systems may be forced to be highly correlated and the impostordistributions to be lightly correlated. The correlation coefficients (ρ)are shown in Table 2. The subscripts denote the connection. A plot ofthe joint pdf for the fusion of system #1 with system #2 is shown inFIG. 7, where the correlation between the authentic distributions isquite evident.

The single biometric ROC curves are shown in FIG. 8. As could bepredicted from the pdf curves plotted in FIG. 6, System #1 performs muchbetter than the other two systems, with System #3 having the worstperformance.

Fusing 2 systems at a time; there are three possibilities: #1+#2, #1+#3,and #2+#3. The resulting ROC curves are shown in FIG. 9. As predicted bythe corollary, each 2-system pair outperforms their individualcomponents. Although the fusion of system #2 with system #3 has worseperformance than system #1 alone, it is still better than the singlesystem performance of either system #2 or system #3.

Finally, FIG. 10 depicts the results when fusing all three systems andcomparing its performance with the performance of the 2-system pairs.The addition of the third biometric system gives substantial improvementover the best performing pair of biometrics.

Tests were conducted on individual and fused biometric systems in orderto determine whether the theory presented above accurately predicts whatwill happen in a real-world situation. The performance of threebiometric systems were considered. The numbers of score samplesavailable are listed in Table 3. The scores for each modality werecollected independently from essentially disjoint subsets of the generalpopulation.

TABLE 3 Number of Authentic Biometric Number of Impostor Scores ScoresFingerprint 8,500,000 21,563 Signature 325,710 990 Facial Recognition4,441,659 1,347

To simulate the situation of an individual obtaining a score from eachbiometric, the initial thought was to build a “virtual” match from thedata of Table 3. Assuming independence between the biometrics, a 3-tupleset of data was constructed. The 3-tuple set was an ordered set of threescore values, by arbitrarily assigning a fingerprint score and a facialrecognition score to each signature score for a total of 990 authenticscore 3-tuples and 325,710 impostor score 3-tuples.

By assuming independence, it is well known that the joint pdf is theproduct of the marginal density functions, hence the joint classconditional pdf for the three biometric systems, ƒ(x|θ)=ƒ(x₁,x₂,x₃|θ)can be written asƒ(x|θ)=ƒ_(fingerprint)(x|θ)ƒ_(signature)(x|θ)ƒ_(facial)(x|θ)  (33)

So it is not necessary to dilute the available data. It is sufficient toapproximate the appropriate marginal density functions for each modalityusing all the data available, and compute the joint pdf using Equation33.

The class condition density functions for each of the three modalities,ƒ_(fingerprint)(x|θ), ƒ_(signature)(x|θ) and ƒ_(facial)(x|θ), wereestimated using the available sampled data. The authentic scoredensities were approximated using two methods: (1) thehistogram-interpolation technique and (2) the Parzen-window method. Theimpostor densities were approximated using the histogram-interpolationmethod. Although each of these estimation methods are guaranteed toconverge in probability to the true underlying density as the number ofsamples goes to infinity, they are still approximations and canintroduce error into the decision making process, as will be seen in thenext section. The densities for fingerprint, signature, and facialrecognition are shown in FIG. 11, FIG. 12 and FIG. 13 respectively. Notethat the authentic pdf for facial recognition is bimodal. TheNeyman-Pearson test was used to determine the optimal ROC for each ofthe modalities. The ROC curves for fingerprint, signature, and facialrecognition are plotted in FIG. 14.

There are three possible unique pairings of the three biometric systems:(1) fingerprints with signature, (2) fingerprints with facialrecognition, and (3) signatures with facial recognition. Using themarginal densities (above) to create the required 2-D joint classconditional density functions, two sets of 2-D joint density functionswere computed—one in which the authentic marginal densities wereapproximated using the histogram method, and one in which the densitieswere approximated using the Parzen window method. Using theNeyman-Pearson test, an optimal ROC was computed for each fused pairingand each approximation method. The ROC curves for the histogram methodare shown in FIG. 15 and the ROC curves for the Parzen window method areshown in FIG. 16.

As predicted by the corollary, the fused performance is better than theindividual performance for each pair under each approximation method.But, as we cautioned in the example, error due to small sample sizes cancause pdf distortion. This is apparent when fusing fingerprints withsignature data (see FIG. 15 and FIG. 16). Notice that the Parzen-windowROC curve (FIG. 16) crosses over the curve forfingerprint-facial-recognition fusion, but does not cross over whenusing the histogram interpolation method (FIG. 15). Small differencesbetween the two sets of marginal densities are magnified when usingtheir product to compute the 2-dimensional joint densities, which isreflected in the ROC.

As a final step, all three modalities were fused. The resulting ROCusing histogram interpolating is shown in FIG. 17, and the ROC using theParzen window is shown in FIG. 18. As might be expected, the Parzenwindow pdf distortion with the 2-dimensional fingerprint-signature casehas carried through to the 3-dimensional case. The overall performance,however, is dramatically better than any of the 2-dimensionalconfigurations as predicted by the corollary.

In the material presented above, it was assumed that all N biometricscores would be available for fusion. Indeed, the Neyman-Pearson Lemmaguarantees that this provides the most powerful test for a fixed FAR. Inpractice, however, this could be an expensive way of doing business. Ifit is assumed that all aspects of using a biometric system, time, risk,etc., can be equated to a cost, then a cost function can be constructed.Below, we construct the cost function and show how it can be minimizedusing the Neyman-Pearson test. Hence, the Neyman-Pearson theory is notlimited to “all-at-once” fusion—it can be used for serial, parallel, andhierarchal systems.

In the following section, a review of the development of the costfunction for a verification system is provided, and then a cost functionfor an identification system is developed. For each system, an algorithmis presented that uses the Neyman-Pearson test to minimize the costfunction for a second order modality biometric system. The cost functionis presented for the general case of N-biometrics. Because minimizingthe cost function is recursive, the computational load growsexponentially with added dimensions. Hence, an efficient algorithm isneeded to handle the general case.

From the prior discussion of the cost function for a 2-station system,the costs for using the first biometric, the second biometric, and themanual check are c₁,c₂, and c₃, respectively, and FAR_(sys) isspecified. The first-station test cannot have a false-acceptance-rate,FAR₁, that exceeds FAR_(sys). Given a test with a specified FAR₁, thereis an associated false-rejection-rate, FRR₁, which is the fraction ofsubjects that, on averages are required to move on to the secondstation. The FAR required at the second station is FAR₂=FAR_(sys)−FAR₁.It is known that P(A∪B)=P(A)+P(B)−P(A)P(B), so the computation of FAR₂appears imperfect. However, if FAR_(sys)=P(A∪B) and FAR₁=P(A), in theconstruction of the decision space, it is intended thatFAR₂=P(B)−P(A)P(B).

Given a test with the computed FAR₂, there is an associatedfalse-rejection-rate, FRR₂ by the second station, which is the fractionof subjects that are required to move on to a manual checkout station.This is all captured by the following cost functionCost=c ₁+FRR₁(c ₂ +c ₃FRR₂)  (30)

There is a cost for every choice of FAR₁≦FAR_(sys), so the Cost inEquation 30 is a function of FAR₁. For a given test method, there existsa value of FAR₁ that yields the smallest cost and we develop analgorithm to find that value.

Using a modified Neyman-Pearson test to optimally minimize Equation 30,an algorithm can be derived. Step 1: set the initial cost estimate toinfinity and, for a specified FAR_(sys), loop over all possible valuesof FAR₁≦FAR_(sys). To be practical, in the algorithm a finite sample ofthe infinite possible values may be used. The first step in the loop isto use the Neyman-Pearson test at FAR₁ to determine the most powerfultest, CAR₁, for the first biometric, and FRR₁=1−CAR₁ is computed. Sinceit is one dimensional, the critical region is a collection of disjointintervals I^(Au). As in the proof to the corollary, the I-dimensionalI_(Au) is recast as a 2-dimensional region, R₁=I_(Au)×γ₂⊂R², so that

$\begin{matrix}{{CAR}_{I} = {{\int_{R_{I}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}} = {\int_{I_{Au}}^{\;}{{f\left( x_{I} \middle| {Au} \right)}{\mathbb{d}x}}}}} & (34)\end{matrix}$

When it is necessary to test against the second biometric, the region R₁of the decision space is no longer available since the first test usedit up. The Neyman-Pearson test can be applied to the reduced decisionspace, which is the compliment of R₁. Step 2: FAR₂=FAR_(sys)−FAR₁ iscomputed. Step 3: FAR₂ is used in the Neyman-Pearson test to determinethe most powerful test, CAR₂, for the second biometric fused with thefirst biometric over the reduced decision space R₁ ^(C). The criticalregion for CAR₂ is R₂, which is disjoint from R₁ by the construction.Score pairs that result in the failure to be authenticated at eitherbiometric station must fall within the region R₃=(R₁∪R₂)^(C), from whichit is shown that

${FRR}_{2} = \frac{\int_{R_{3}}^{\;}{{f\left( x \middle| {Au} \right)}{\mathbb{d}x}}}{{FRR}_{1}}$

Step 4: compute the cost at the current setting of FAR₁ using FRR₁ andFRR₂. Step 5: reset the minimum cost if cheaper, and keep track of theFAR₁ responsible for the minimum cost. A typical cost function is shownin FIG. 19.

Two special cases should be noted. In the first case, if c₁>0, c₂>0, andc₃=0, the algorithm shows that the minimum cost is to use only the firstbiometric—that is, FAR₁=FAR_(sys). This makes sense because there is nocost penalty for authentication failures to bypass the second stationand go directly to the manual check.

In the second case, c₁=c₂=0 and c₃>0, the algorithm shows that scoresshould be collected from both stations and fused all at once; that is,FAR₁=1.0. Again, this makes sense because there is no cost penalty forcollecting scores at both stations, and because the Neyman-Pearson testgives the most powerful CAR (smallest FRR) when it can fuse both scoresat once.

To extend the cost function to higher dimensions, the logic discussedabove is simply repeated to arrive at

$\begin{matrix}{{Cost} = {c_{1} + {c_{2}{FRR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}{FRR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}{FRR}_{i}}}}} & (35)\end{matrix}$In minimizing Equation 35, there is 1 degree of freedom, namely FAR₁.Equation 35 has N−1 degrees of freedom. When FAR₁≦FAR_(sys) are set,then FAR₂≦FAR₁ can bet set, then FAR₃≦FAR₂, and so on—and to minimizethus, N−1 levels of recursion.

Identification System: Identification (one-to-many) systems arediscussed. With this construction, each station attempts to discardimpostors. Candidates that cannot be discarded are passed on to the nextstation. Candidates that cannot be discarded by the biometric systemsarrive at the manual checkout. The goal, of course, is to prune thenumber of impostor templates thus limiting the number that move on tosubsequent steps. This is a logical AND process—for an authentic matchto be accepted, a candidate must pass the test at station 1 and station2 and station 3 and so forth. In contrast to a verification system, thesystem administrator must specify a system false-rejection-rate,FRR_(sys) instead of a FAR_(sys). But just like the verification systemproblem the sum of the component FRR values at each decision pointcannot exceed FRR_(sys). If FRR with FAR are replaced in Equations 34 or35 the following cost equations are arrived at for 2-biometric and anN-biometric identification system

$\begin{matrix}{{Cost} = {c_{1} + {{FAR}_{1}\left( {c_{2} + {c_{3}{FAR}_{2}}} \right)}}} & (36) \\{{Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}{FAR}_{i}}}}} & (37)\end{matrix}$Alternately, equation 37 may be written as:

$\begin{matrix}{{{Cost} = {\sum\limits_{j = 0}^{N}\left( {c_{j + 1}{\prod\limits_{i = 1}^{j}{FAR}_{i}}} \right)}},} & \left( {37A} \right)\end{matrix}$These equations are a mathematical dual of Equations 34 and 35 are thusminimized using the logic of the algorithm that minimizes theverification system.

Algorithm: Generating Matching and Non-Matching PDF Surfaces. Theoptimal fusion algorithm uses the probability of an authentic match andthe probability of a false match for each p_(ij)εP. These probabilitiesmay be arrived at by numerical integration of the sampled surface of thejoint pdf. A sufficient number of samples may be generated to get a“smooth” surface by simulation. Given a sequence of matching score pairsand non-matching score pairs, it is possible to construct a numericalmodel of the marginal cumulative distribution functions (cdf). Thedistribution functions may be used to generate pseudo random scorepairs. If the marginal densities are independent, then it isstraightforward to generate sample score pairs independently from eachcdf. If the densities are correlated, we generate the covariance matrixand then use Cholesky factorization to obtain a matrix that transformsindependent random deviates into samples that are appropriatelycorrelated.

Assume a given partition P. The joint pdf for both the authentic andimpostor cases may be built by mapping simulated score pairs to theappropriate P_(ij)εP and incrementing a counter for that sub-square.That is, it is possible to build a 2-dimensional histogram, which isstored in a 2-dimensional array of appropriate dimensions for thepartition. If we divide each array element by the total number ofsamples, we have an approximation to the probability of a score pairfalling within the associated sub-square. We call this type of an arraythe probability partition array (PPA). Let P_(fm) be the PPA for thejoint false match distribution and let P_(m) be the PPA for theauthentic match distribution. Then, the probability of an impostor'sscore pair, (s₁,s₂)εP_(ij), resulting in a match is P_(fm)(i,j).Likewise, the probability of a score pair resulting in a match when itshould be a match is P_(m)(i,j). The PPA for a false reject (does notmatch when it should) is P_(fr)=1−P_(m).

Consider the partition arrays, P_(fm)and P_(m), defined in Section #5.Consider the ratio

$\frac{P_{fm}\left( {i,j} \right)}{P_{m}\left( {i,j} \right)},{P_{fr} > 0.}$The larger this ratio the more the sub-square indexed by (i,j) favors afalse match over a false reject. Based on the propositions presented inSection 4, it is optimal to tag the sub-square with the largest ratio aspart of the no-match zone, and then tag the next largest, and so on.

Therefore, an algorithm that is in keeping with the above may proceedas:

-   Step 1. Assume a required FAR has been given.-   Step 2. Allocate the array P_(mz) to have the same dimensions as    P_(m), and P_(fm). This is the match zone array. Initialize all of    its elements as belonging to the match zone.-   Step 3. Let the index set A={all the (i,j) indices for the    probability partition arrays}.-   Step 4. Identify all indices in A such that P_(m)(i,j)=P_(fm)(i,j)=0    and store those indices in the indexing set Z={(i,j):    -   P_(m)(i,j)=P_(fm)(i,j)=0}. Tag each element in P_(mz) indexed by        Z as part of the no-match zone.-   Step 5. Let B=A−Z={(i,j): P_(m)(i,j)≠0 and P_(fm)(i,j)≠0}. We    process only those indices in B. This does not affect optimality    because there is most likely zero probability that either a false    match score pair or a false reject score pair falls into any    sub-square indexed by elements of Z.-   Step 6. Identify all indices in B such that    -   P_(fm)(i,j)>0 and P_(m)(i,j)=0 and store those indices in        Z_(∞)={(i,j)_(k):    -   P_(fm)(i,j)>0, P_(m)(i,j)=0}. Simply put, this index set        includes the indexes to all the sub-squares that have zero        probability of a match but non-zero probability of false match.-   Step 7. Tag all the sub-squares in P_(mz) indexed by Z_(∞) as    belonging to the no-match zone. At this point, the probability of a    matching score pair falling into the no-match zone is zero, The    probability of a non-matching score pair falling into the match zone    is:    FAR_(Z) _(∞) =1−Σ_((i,j)εZ) _(∞) P _(fm)(i,j)    Furthermore, if FAR_(Z) _(∞) <=FAR then we are done and can exit the    algorithm.-   Step 8 Otherwise, we construct a new index set

$\quad\begin{matrix}{C = {A - Z - Z_{\infty}}} \\{= {\left\{ {{{\left( {i,j} \right)_{k}\text{:}\mspace{14mu}{P_{fm}\left( {i,j} \right)}_{k}} > 0},{{P_{m}\left( {i,j} \right)}_{k} > 0},{\frac{{P_{fm}\left( {i,j} \right)}_{k}}{{P_{m}\left( {i,j} \right)}_{k}}>=\frac{{P_{fm}\left( {i,j} \right)}_{k + 1}}{{P_{m}\left( {i,j} \right)}_{k + 1}}}} \right\}.}}\end{matrix}$We see that C holds the indices of the non-zero probabilities in asorted order—the ratios of false-match to match probabilities occur indescending order.

-   Step 9. Let C_(N) be the index set that contains the first N indices    in C. We determine N so that:    FAR_(Z) _(∞) _(∪C) _(N) =1−Σ_((i,j)εZ) _(∞) P _(fm)(i,j)−Σ_((i,j)εC)    _(N) P _(fm)(i,j)≦FAR-   Step 10. Label elements of P_(mz) indexed by members of C_(N) as    belonging to the no-match zone. This results in a FRR given by    FRR=Σ_((i,j)εC) _(N) P _(m)(i,j),    and furthermore this FRR is optimal.

It will be recognized that other variations of these steps may be made,and still be within the scope of the invention. For clarity, thenotation “(i,j)” is used to identify arrays that have at least twomodalities. Therefore, the notation “(i,j)” includes more than twomodalities, for example (i,j,k), (i,j,k,l), (i,j,k,l,m), etc.

For example, FIG. 20 illustrates one method according to the inventionin which Pm(i,j) is provided 310 and Pfm(i,j) is provided 313. As partof identifying 316 indices (i,j) corresponding to a no-match zone, afirst index set (“A”) may be identified. The indices in set A may be the(i,j) indices that have values in both Pfm(i,j) and Pm(i,j). A secondindex set (“Z∞”) may be identified, the indices of Z∞ being the (i,j)indices in set A where both Pfm(i,j) is larger than zero and Pm(i,j) isequal to zero. Then determine FAR_(Z∞) 319, where FAR_(Z) _(∞)=1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j). It should be noted that the indices ofZ∞ may be the indices in set A where Pm(i,j) is equal to zero, since theindices where both Pfm(i,j)=Pm(i,j)=0 will not affect FAR_(Z∞), andsince there will be no negative values in the probability partitionarrays. However, since defining the indices of Z∞ to be the (i,j)indices in set A where both Pfm(i,j) is larger than zero and Pm(i,j) isequal to zero yields the smallest number of indices for Z_(∞), we willuse that definition for illustration purposes since it is the least thatmust be done for the mathematics of FAR_(Z∞) to work correctly. It willbe understood that larger no-match zones may be defined, but they willinclude Z∞, so we illustrate the method using the smaller Z∞ definitionbelieving that definitions of no-match zones that include Z∞ will fallwithin the scope of the method described.

FAR_(Z∞) may be compared 322 to a desired false-acceptance-rate (“FAR”),and if FAR_(Z∞) is less than or equal to the desiredfalse-acceptance-rate, then the data set may be accepted, iffalse-rejection-rate is not important. If FAR_(Z∞) is greater than thedesired false-acceptance-rate, then the data set may be rejected 325.

If false-rejection-rate is important to determining whether a data setis acceptable, then indices in a match zone may be selected, ordered,and some of the indices may be selected for further calculations 328.Toward that end, the following steps may be carried out. The method mayfarther include identifying a third index set ZM∞, which may be thoughtof as a modified Z∞, that is to say a modified no-match zone. Here wemodify Z∞ so that ZM∞ includes indices that would not affect thecalculation for FAR_(Z∞), but which might affect calculations related tothe false-rejection-rate. The indices of ZM∞ may be the (i,j) indices inZ∞ plus those indices where both Pfm(i,j) and Pm(i,j) are equal to zero.The indices where Pfm(i,j)=Pm(i,j)=0 are added to the no-match zonebecause in the calculation of a fourth index set (“C”), these indicesshould be removed from consideration. The indices of C may be the (i,j)indices that are in A but not ZM∞. The indices of C may be arranged suchthat

$\frac{{P_{fm}\left( {i,j} \right)}_{k}}{{P_{m}\left( {i,j} \right)}_{k}}>=\frac{{P_{fm}\left( {i,j} \right)}_{k + 1}}{{P_{m}\left( {i,j} \right)}_{k + 1}}$to provide an arranged C index. A fifth index set (“Cn”) may beidentified. The indices of Cn may be the first N (i,j) indices of thearranged C index, where N is a number for which the following is true:FAR_(Z) _(∞) _(∪C) _(N) =1−Σ_((i,j)εZ) _(∞) P _(fm)(i,j)−Σ_((i,j)εC)_(N) P _(fm)(i,j)≦FAR.

The FRR may be determined 331, where FRR=Σ_((i,j)εC) _(N) P_(m)(i,j),and compared 334 to a desired false-rejection-rate. If FRR is greaterthan the desired false-rejection-rate, then the data set may be rejected337, even though FAR_(Z∞) is less than or equal to the desiredfalse-acceptance-rate. Otherwise, the data set may be accepted.

FIG. 21 illustrates another method according to the invention in whichthe FRR may be calculated first. In FIG. 21, the reference numbers fromFIG. 20 are used, but some of the steps in FIG. 21 may vary somewhatfrom those described above. In such a method, a first index set (“A”)may be identified, the indices in A being the (i,j) indices that havevalues in both Pfm(i,j) and Pm(i,j). A second index set (“Z∞”), which isthe no match zone, may be identified 316. The indices of Z∞ may be the(i,j) indices of A where Pm(i,j) is equal to zero. Here we use the moreinclusive definition for the no-match zone because the calculation ofset C comes earlier in the process. A third index set (“C”) may beidentified, the indices of C being the (i,j) indices that are in A butnot Z∞. The indices of C may be arranged such that

$\frac{{P_{fm}\left( {i,j} \right)}_{k}}{{P_{m}\left( {i,j} \right)}_{k}}>=\frac{{P_{fm}\left( {i,j} \right)}_{k + 1}}{{P_{m}\left( {i,j} \right)}_{k + 1}}$to provide an arranged C index, and a fourth index set (“Cn”) may beidentified. The indices of Cn may be the first N (i,j) indices of thearranged C index, where N is a number for which the following is true:FAR_(Z) _(∞) _(∪C) _(N) =1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j)−Σ_((i,j)εC) _(N)P_(fm)(i,j)≦FAR. The FRR may be determined, where FRR=Σ_((i,j)εC) _(N)P_(m)(i,j), and compared to a desired false-rejection-rate. If the FRRis greater than the desired false-rejection-rate, then the data set maybe rejected. If the FRR is less than or equal to the desiredfalse-rejection-rate, then the data set may be accepted, iffalse-acceptance-rate is not important. If false-acceptance-rate isimportant, the FAR_(Z∞) may be determined, where FAR_(Z) _(∞)=1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j). Since the more inclusive definition isused for Z∞ in this method, and that more inclusive definition does notaffect the value of FAR_(Z∞), we need not add back the indices whereboth Pfm(i,j)=Pm(i,j)=0. The FAR_(Z∞) may be compared to a desiredfalse-acceptance-rate, and if FAR_(Z∞) in is greater than the desiredfalse-acceptance-rate, then the data set may be rejected. Otherwise, thedata set may be accepted.

It may now be recognized that a simplified form of a method according tothe invention may be executed as follows.

-   step 1: provide a first probability partition array (“Pm(i,j)”), the    Pm(i,j) being comprised of probability values for information pieces    in the data set, each probability value in the Pm(i,j) corresponding    to the probability of an authentic match;-   step 2; provide a second probability partition array (“Pfm(i,j)”),    the Pfm(i,j) being comprised of probability values for information    pieces in the data set, each probability value in the Pfm(i,j)    corresponding to the probability of a false match;-   step 3: identify a first index set (“A”), the indices in set A being    the (i,j) indices that have values in both Pfm(i,j) and Pm(i,j);-   step 4: execute at least one of the following;    -   (a) identify a first no-match zone (“Z1∞”) that includes at        least the indices of set A for which both Pfm(i,j) is larger        than zero and Pm(i,j) is equal to zero, and use Z1∞ to determine        FAR_(Z∞), where FAR_(Z) _(∞) =1−Σ_((i,j)εZ) _(∞) P_(fm)(i,j),        and compare FAR_(Z∞) to a desired false-acceptance-rate, and if        FAR_(Z∞) is greater than the desired false-acceptance-rate, then        reject the data set;    -   (1) identify a second no-match zone (“Z2∞”) that includes the        indices of set A for which Pm(i,j) is equal to zero, and use Z2∞        to identify a second index set (“C”), the indices of C being the        (i,j) indices that are in A but not Z2∞, and arrange the (i,j)        indices of C such that

$\frac{{P_{fm}\left( {i,j} \right)}_{k}}{{P_{m}\left( {i,j} \right)}_{k}}>=\frac{{P_{fm}\left( {i,j} \right)}_{k + 1}}{{P_{m}\left( {i,j} \right)}_{k + 1}}$to provide an arranged C index, and identify a third index set (“Cn”),the indices of Cn being the first N (i,j) indices of the arranged Cindex, where N is a number for which the following is true:

${FAR}_{Z_{x} \Cup C_{N}} = {{1 - {\sum\limits_{{({i,j})} \in Z_{x}}{P_{f\; m}\left( {i,j} \right)}} - {\sum\limits_{{({i,j})} \in C_{N}}{P_{f\; m}\left( {i,j} \right)}}} \leq {FAR}}$${\text{and determine~~}{FRR}},{{\text{~~where}\mspace{14mu}{FDD}} = {\sum\limits_{{({i,j})} \in C_{N}}{P_{m}\left( {i,j} \right)}}},$and compare FRR to a desired false-rejection-rate, and if FRR is greaterthan the desired false-rejection-rate, then reject the data set.If FAR_(Z∞) is less than or equal to the desired false-acceptance-rate,and FRR is less than or equal to the desired false-rejection-rate, thenthe data set may be accepted. Z1∞ may be expanded to include additionalindices of set A by including in Z1∞ the indices of A for which Pm(i,j)is equal to zero. In this manner, a single no-match zone may be definedand used for the entire step 4.

Although the present invention has been described with respect to one ormore particular embodiments, it will be understood that otherembodiments of the present invention may be made without departing fromthe spirit and scope of the present invention. Hence, the presentinvention is deemed limited only by the appended claims and thereasonable interpretation thereof.

1. A method of authorizing a transaction, comprising: enrolling at leasttwo biometric specimens, a first one of the specimens being a firsttype, and a second one of the specimens being a second type; determininga false acceptance ratio (“FAR”) for each of the specimens, wherein atleast one of the false acceptance ratios is determined using thefollowing equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au)) f(x|Im)𝕕x; identifyingauthorization options, each option requiring a match to one or more ofthe biometric specimens; calculating a cost value for at least one ofthe options to provide a calculated cost value (“CCV”), wherein the CCVis a function of the FAR(s) of the specimen(s) corresponding to theoption; identifying an acceptable cost value range (“ACV range”);comparing the CCV to the ACV range; determining whether the CCV is inthe ACV range; selecting the option if the CCV is in the ACV range;providing a set of biometric sample(s) (the “sample set”), the sampleset having biometric sample(s) of the same type(s) as thosecorresponding to the selected option; comparing the biometric sample(s)to the biometric specimen(s); determining whether the biometricsample(s) match the biometric specimen(s); and if the biometricsample(s) match the biometric specimen(s), then authorizing thetransaction.
 2. The method of claim 1, wherein the types are selectedfrom fingerprint, facial image, retinal image, iris scan, hand geometryscan, voice print, signature, signature gait, keystroke tempo, bloodvessel pattern, palm print, skin composition spectrum, lip shape and earshape.
 3. The method of claim 1, wherein a first type corresponds to afirst body part and a second type corresponds to a second body part. 4.The method of claim 1, wherein the biometric specimens comprise a set,and the options comprise permutations of the set.
 5. The method of claim1, wherein at least one of the cost values is calculated using thefollowing equation${Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}{{FAR}_{i}.}}}}$6. The method of claim 1, wherein the ACV range is provided by anadministrator.
 7. A computer readable memory device having storedthereon instructions that are executable by a computer to cause thecomputer to: enroll at least two biometric specimens, a first one of thespecimens being a first type, and a second one of the specimens being asecond type; determine a false acceptance ratio (“FAR”) for each of thespecimens, wherein the instructions cause the computer to determine atleast one of the false acceptance ratios using the following equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au)) f(x|Im)𝕕x; identifyauthorization options, each option requiring a match to one or more ofthe biometric specimens; calculate a cost value for at least one of theoptions to provide a calculated cost value (“CCV”), wherein the CCV is afunction of the FAR(s) of the specimen(s) corresponding to the option;identify an acceptable cost value range (“ACV range”); compare the CCVto the ACV range; determine whether the CCV is in the ACV range; selectthe option if the CCV is in the ACV range; provide a set of biometricsample(s) (the “sample set”), the sample set having biometric sample(s)of the same type(s) as those corresponding to the selected option;compare the biometric sample(s) to the biometric specimen(s); determinewhether the biometric sample(s) match the biometric specimen(s); and ifthe biometric sample(s) match the biometric specimen(s), then authorizethe transaction.
 8. The memory device of claim 7, wherein theinstructions cause the computer to calculate at least one of the costvalues using the following equation${Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}{{FAR}_{i}.}}}}$9. An authorization system, comprising: a computer capable of executingcomputer-readable instructions; at least one biometric specimen readerin communication with the computer; at least one biometric sample readerin communication with the computer; a database; computer-readableinstructions provided to the computer for causing the computer to:enroll in the database at least two biometric specimens via one or moreof the biometric specimen readers, a first one of the specimens being afirst type, and a second one of the specimens being a second type;determine a false acceptance ratio (“FAR”) for each of the specimens,wherein the instructions cause the computer to determine at least one ofthe false acceptance ratios using the following equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au)) f(x|Im)𝕕x; identifyauthorization options, each option requiring a match to one or more ofthe biometric specimens; calculate a cost value for at least one of theoptions to provide a calculated cost value (“CCV”), wherein the CCV is afunction of the FAR(s) of the specimen(s) corresponding to the option;identify an acceptable cost value range (“ACV range”); compare the CCVto the ACV range; determine whether the CCV is in the ACV range; selectthe option if the CCV is in the ACV range; provide a set of biometricsample(s) (the “sample set”) via one or more of the biometric samplereaders, the sample set having biometric sample(s) of the same type(s)as those corresponding to the selected option; compare the biometricsample(s) to the biometric specimen(s); determine whether the biometricsample(s) match the biometric specimen(s); and if the biometricsample(s) match the biometric specimen(s), then authorize thetransaction.
 10. The system of claim 9, wherein the instructions causethe computer to calculate at least one of the cost values using thefollowing equation${Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}{{FAR}_{i}.}}}}$11. A method of authorizing a transaction, comprising: enrolling atleast two biometric specimens, a first one of the specimens being afirst type, and a second one of the specimens being a second type,wherein the biometric specimens comprise a set; determining a falseacceptance ratio (“FAR”) for each of the specimens; identifyingauthorization options, each option requiring a match to one or more ofthe biometric specimens, wherein the options comprise permutations ofthe set of biometric specimens; calculating a cost value for at leastone of the options to provide a calculated cost value (“CCV”), whereinthe CCV is a function of the FAR(s) of the specimen(s) corresponding tothe option; identifying an acceptable cost value range (“ACV range”);comparing the CCV to the ACV range; determining whether the CCV is inthe ACV range; selecting the option if the CCV is in the ACV range;providing a set of biometric sample(s) (the “sample set”), the sampleset having biometric sample(s) of the same type(s) as thosecorresponding to the selected option; comparing the biometric sample(s)to the biometric specimen(s); determining whether the biometricsample(s) match the biometric specimen(s); and if the biometricsample(s) match the biometric specimen(s), then authorizing thetransaction.
 12. The method of claim 11, wherein the types are selectedfrom fingerprint, facial image, retinal image, iris scan, hand geometryscan, voice print, signature, signature gait, keystroke tempo, bloodvessel pattern, palm print, skin composition spectrum, lip shape and earshape.
 13. The method of claim 11, wherein a first type corresponds to afirst body part and a second type corresponds to a second body part. 14.The method of claim 11, wherein at least one of the false acceptanceratios is determined using the following equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au))f(x|Im) 𝕕x.
 15. Themethod of claim 11, wherein at least one of the cost values iscalculated using the following equation${Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\prod\limits_{i = 1}^{2}\;{FAR}_{i}}} + \ldots + {c_{N + 1}{\prod\limits_{i = 1}^{N}\;{{FAR}_{i}.}}}}$16. The method of claim 11, wherein the ACV range is provided by anadministrator.
 17. A method of authorizing a transaction, comprising:enrolling at least two biometric specimens, a first one of the specimensbeing a first type, and a second one of the specimens being a secondtype; determining a false acceptance ratio (“FAR”) for each of thespecimens; identifying authorization options, each option requiring amatch to one or more of the biometric specimens; calculating a costvalue for at least one of the options to provide a calculated cost value(“CCV”), wherein the CCV is a function of the FAR(s) of the specimen(s)corresponding to the option, and at least one of the cost values iscalculated using the following equation${{Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\underset{i = 1}{\prod\limits^{2}}{FAR}_{i}}} + \ldots + {c_{N + 1}{\underset{i = 1}{\prod\limits^{N}}{FAR}_{i}}}}};$identifying an acceptable cost value range (“ACV range”); comparing theCCV to the ACV range; determining whether the CCV is in the ACV range;selecting the option if the CCV is in the ACV range; providing a set ofbiometric sample(s) (the “sample set”), the sample set having biometricsample(s) of the same type(s) as those corresponding to the selectedoption; comparing the biometric sample(s) to the biometric specimen(s);determining whether the biometric sample(s) match the biometricspecimen(s); and if the biometric sample(s) match the biometricspecimen(s), then authorizing the transaction.
 18. The method of claim17, wherein the types are selected from fingerprint, facial image,retinal image, iris scan, hand geometry scan, voice print, signature,signature gait, keystroke tempo, blood vessel pattern, palm print, skincomposition spectrum, lip shape and ear shape.
 19. The method of claim17, wherein a first type corresponds to a first body part and a secondtype corresponds to a second body part.
 20. The method of claim 17,wherein at least one of the false acceptance ratios is determined usingthe following equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au))f(x|Im) 𝕕x.
 21. Themethod of claim 17, wherein the biometric specimens comprise a set, andthe options comprise permutations of the set.
 22. The method of claim17, wherein the ACV range is provided by an administrator.
 23. Acomputer readable memory device having stored thereon instructions thatare executable by a computer to cause the computer to: enroll at leasttwo biometric specimens, a first one of the specimens being a firsttype, and a second one of the specimens being a second type; determine afalse acceptance ratio (“FAR”) for each of the specimens; identifyauthorization options, each option requiring a match to one or more ofthe biometric specimens; calculate a cost value for at least one of theoptions to provide a calculated cost value (“CCV”), wherein the CCV is afunction of the FAR(s) of the specimen(s) corresponding to the option,and the instructions cause the computer to calculate at least one of thecost values using the following equation${{Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\underset{i = 1}{\prod\limits^{2}}{FAR}_{i}}} + \ldots + {c_{N + 1}{\underset{i = 1}{\prod\limits^{N}}{FAR}_{i}}}}};$identify an acceptable cost value range (“ACV range”); compare the CCVto the ACV range; determine whether the CCV is in the ACV range; selectthe option if the CCV is in the ACV range; provide a set of biometricsample(s) (the “sample set”), the sample set having biometric sample(s)of the same type(s) as those corresponding to the selected option;compare the biometric sample(s) to the biometric specimen(s); determinewhether the biometric sample(s) match the biometric specimen(s); and ifthe biometric sample(s) match the biometric specimen(s), then authorizethe transaction.
 24. The memory device of claim 23, wherein theinstructions cause the computer to determine at least one of the falseacceptance ratios using the following equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au))f(x|Im) 𝕕x.
 25. Anauthorization system, comprising: a computer capable of executingcomputer-readable instructions; at least one biometric specimen readerin communication with the computer; at least one biometric sample readerin communication with the computer; a database; computer-readableinstructions provided to the computer for causing the computer to:enroll in the database at least two biometric specimens via one or moreof the biometric specimen readers, a first one of the specimens being afirst type, and a second one of the specimens being a second type;determine a false acceptance ratio (“FAR”) for each of the specimens;identify authorization options, each option requiring a match to one ormore of the biometric specimens; calculate a cost value for at least oneof the options to provide a calculated cost value (“CCV”), wherein theCCV is a function of the FAR(s) of the specimen(s) corresponding to theoption, and the instructions cause the computer to calculate at leastone of the cost values using the following equation${{Cost} = {c_{1} + {c_{2}{FAR}_{1}} + {c_{3}{\underset{i = 1}{\prod\limits^{2}}{FAR}_{i}}} + \ldots + {c_{N + 1}{\underset{i = 1}{\prod\limits^{N}}{FAR}_{i}}}}};$identify an acceptable cost value range (“ACV range”); compare the CCVto the ACV range; determine whether the CCV is in the ACV range; selectthe option if the CCV is in the ACV range; provide a set of biometricsample(s) (the “sample set”) via one or more of the biometric samplereaders, the sample set having biometric sample(s) of the same type(s)as those corresponding to the selected option; compare the biometricsample(s) to the biometric specimen(s); determine whether the biometricsample(s) match the biometric specimen(s); and if the biometricsample(s) match the biometric specimen(s), then authorize thetransaction.
 26. The system of claim 25, wherein the instructions causethe computer to determine at least one of the false acceptance ratiosusing the following equationFAR = P_(Type  I) = P(x ∈ R_(Au)|Im) = ∫_(R_(Au))f(x|Im) 𝕕x.